\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 30, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2007/30\hfil Subparabolic comparison principle]
{A comparison principle for a class of subparabolic equations in
Grushin-type spaces}
\author[T. Bieske\hfil EJDE-2007/30\hfilneg]
{Thomas Bieske}
\address{Thomas Bieske \newline
Department of Mathematics\\
University of South Florida\\
Tampa, FL 33620, USA}
\email{tbieske@math.usf.edu}
\thanks{Submitted November 27, 2006. Published February 14, 2007.}
\subjclass[2000]{35K55, 49L25, 53C17}
\keywords{Grushin-type spaces; parabolic equations; viscosity solutions}
\begin{abstract}
We define two notions of viscosity solutions to subparabolic
equations in Grushin-type spaces, depending on whether the test
functions concern only the past or both the past and the future.
We then prove a comparison principle for a class of subparabolic
equations and show the sufficiency of considering the test functions
that concern only the past.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\section{Background and Motivation}
In \cite{B:GS}, the author considered viscosity solutions to fully
nonlinear subelliptic equations in Grushin-type spaces, which are
sub-Riemannian metric spaces lacking a group structure. It is
natural to consider viscosity solutions to subparabolic equations
in this same environment. Our main theorem, found in Section 4, is
a comparison principle for a class of subparabolic equations in
Grushin-type spaces. We begin with a short review of the key
geometric properties of Grushin-type spaces in Section 2 and in
Section 3, we define two notions of viscosity solutions to
subparabolic equations. Section 4 contains a parabolic comparison
principle and the corollary showing the sufficiency of using test
functions that concern only the past.
\section{Grushin-type Spaces}
We begin with $\mathbb{R}^n$, possessing coordinates $p =
(x_1,x_2,\dots,x_n)$ and vector fields
$$
X_i = \rho_i (x_1,x_2,\dots,x_{i-1}) \frac{\partial}{\partial x_i}
$$
for $i=2,3,\dots, n$ where $\rho_i (x_1,x_2,\dots,x_{i-1})$ is a
(possibly constant) polynomial. We decree that $\rho_1 \equiv 1$
so that
$$
X_1=\frac{\partial}{\partial x_1}.
$$
A quick calculation shows that when $i < j$, the Lie bracket is given by
\begin{equation*}
X_{ij} \equiv [X_i,X_j]= \rho_i (x_1,x_2,\dots,x_{i-1})
\frac{\partial \rho_j (x_1,x_2,\dots,x_{j-1})}{\partial x_i}
\frac{\partial }{\partial x_j}.
\end{equation*}
Because the $\rho_i$'s are polynomials, at each point there is a finite
number of iterations of the Lie bracket so that
$\frac{\partial}{\partial x_i}$ has
a non-zero coefficient. It follows that
H\"{o}rmander's condition \cite{H:H} is satisfied by these vector fields.
We may further endow $\mathbb{R}^N$ with an inner product
(singular where the polynomials vanish) so that the span of the
$\{X_i\}$ forms an orthonormal basis. This produces a
sub-Riemannian manifold that we shall call $g_n$, which is also
the tangent space to a generalized Grushin-type space $G_n$.
Points in $G_n$ will also be denoted by $p=(x_1,x_2,\dots, x_n)$.
We observe that if $\rho_i\equiv 1$ for all $i$, then
$g_n=G_n=\mathbb{R}^n$.
Given a smooth function $f$ on $G_n$, we define the horizontal gradient
of $f$ as
$$
\nabla_0f(p) = (X_1f(p),X_2f(p),\dots, X_nf(p))
$$
and the symmetrized second order (horizontal) derivative matrix by
$$
((D^2f(p))^{\star})_{ij} = \frac{1}{2} (X_iX_jf(p)+X_jX_if(p))
$$
for $i,j=1,2,\dots n$.
\begin{definition} \label{def1} \rm
The function $f: G_n \to \mathbb{R}$ is said to be
$C^1_{\rm sub}$ if $X_if$ is continuous for all
$i=1,2,\dots,n$. Similarly, the function $f$ is
$C^2_{\rm sub}$ if $X_iX_jf(p)$ is continuous for all
$i,j=1,2,\dots,n$.
\end{definition}
Though $G_n$ is not a Lie group, it is a metric space with the natural
metric being the Carnot-Carath\'{e}odory distance, which is defined for
points $p$ and $q$ as follows:
\begin{equation*}
d_C(p,q)= \inf_{\Gamma} \int_{0}^{1} \| \gamma '(t) \| dt.
\end{equation*}
Here $ \Gamma $ is the set of all curves $ \gamma $ such
that $ \gamma (0) = p$, $\gamma (1) = q $ and
$$
\gamma '(t) \in \mathop{\rm span} \{\{X_i(\gamma(t))\}_{i=1}^n\} .
$$
By Chow's theorem (see, for example, \cite{BR:SRG}) any two points can
be joined by such a curve, which means $ d_C(p,q) $ is an honest
metric. Using this metric, we can define Carnot-Carath\'{e}odory
balls and bounded domains in the usual way.
The Carnot-Carath\'{e}odory metric behaves differently at points
where the polynomials $\rho_i$ vanish. Fixing a point $p_0$,
consider the $n$-tuple
$r_{p_0}=(r^1_{p_0},r^2_{p_0},\dots,r^n_{p_0})$ where $r^i_{p_0}$
is the minimal number of Lie bracket iterations required to
produce
$$
[X_{j_1},[X_{j_2},[\cdots[X_{j_{r^i_{p_0}}},X_i]\cdots](p_0) \neq 0.
$$
Note that though the minimal length is unique, the iteration used to
obtain that minimum is not. Note also that
$$
\rho_i(p_0) \neq 0 \leftrightarrow r^i_{p_0}=0.
$$
Setting $R^i(p_0)=1+r^i_{p_0}$ we obtain the local estimate at
$p_0$
\begin{equation} \label{distest}
d_C(p_0,p) \sim \sum _{i=1}^n |x_i-x_i^0|^\frac{1}{R^i(p_0)}
\end{equation}
as a consequence of \cite[Theorem 7.34]{BR:SRG}. Using this local
estimate, we can construct a local smooth Grushin gauge at the
point $p_0$, denoted $\mathcal{N}(p_0,p)$, that is comparable to
the Carnot-Carath\'{e}odory metric. Namely,
\begin{equation}\label{gauge}
(\mathcal{N}(p_0,p))^{2\mathcal{R}}
=\sum_{i=1}^n (x_i-x_i^0)^{\frac{2\mathcal{R}}{R^i(p_0)}}
\end{equation}
with
$$
\mathcal{R}(p_0)=\prod_{i=1}^n R^i(p_0).
$$
\section{Subparabolic Jets and Solutions to Subparabolic Equations}
In this section, we define and compare various notions of
solutions to parabolic equations in Grushin-type spaces, in the
spirit of \cite[Section 8]{CIL:UGTVS}. We begin by letting
$u(p,t)$ be a function in $G_n \times [0,T]$ for some $T>0$ and by
denoting the set of $n \times n$ symmetric matrices by $S^{n}$. We
consider parabolic equations of the form
\begin{equation}\label{main} u_t+F(t,p,u,\nabla_0
u,(D^2u)^{\star})=0
\end{equation}
for continuous and proper
$F:[0,T]\times G_n \times \mathbb{R} \times g_n \times S^{n} \to \mathbb{R}$.
Recall that $F$ is proper means
$$
F(t,p,r,\eta,X)\leq F(t,p,s,\eta,Y)
$$
when $r\leq s$ and $Y\leq X$ in the usual ordering of symmetric matrices.
\cite{CIL:UGTVS}
We note that the derivatives $\nabla_0 u$ and $(D^2u)^{\star}$
are taken in the space variable $p$. We call such equations
\emph{subparabolic}. Examples of subparabolic equations include
the subparabolic $P$-Laplace equation for $2 \leq P < \infty$
given by
$$
u_t+\Delta_Pu = u_t - \textmd{div}(\|\nabla_0u\|^{P-2}\nabla_0u)=0
$$
and the subparabolic infinite Laplace equation
$$
u_t+\Delta_{\infty}u = u_t - \langle(D^2u)^\star\nabla_0u, \nabla_0u \rangle =0.
$$
Let $\mathcal{O}\subset G_n$ be an open set containing the point $p_0$.
We define the parabolic set $\mathcal{O}_T \equiv \mathcal{O} \times (0,T)$.
Following the definition of Grushin jets in \cite{B:GS}, we can
define the subparabolic superjet of $u(p,t)$ at the point
$(p_0,t_0) \in \mathcal{O}_T$, denoted $P^{2,+}u(p_0,t_0)$, by
using triples $(a,\eta,X) \in \mathbb{R} \times g_n \times S^{n}$
with $\eta=\sum_{i=1}^n \eta_jX_j$ and the $ij$-th entry of $X$
denoted $X_{ij}$. We then have that $(a,\eta,X) \in
P^{2,+}u(p_0,t_0)$ if
\begin{align*}
u(p,t)
&\leq u(p_0,t_0)+ a(t-t_0)+\sum_{j \notin \mathcal{N}}
\frac{1}{\rho_j(p_0)}(x_j-x_j^0)\eta_j\\
&\quad +\frac{1}{2}\sum_{j \notin \mathcal{N}}
\frac{1}{(\rho_j(p_0))^2}(x_j-x_j^0)^2X_{jj} \\
& \quad + \sum_{\stackrel{i,j \notin \mathcal{N}}{i <
j}}(x_i-x_i^0)(x_j-x_j^0)\big(\frac{1}{\rho_j(p_0)\rho_i(p_0)}X_{ij}-
\frac{1}{2}\frac{1}{(\rho_j(p_0))^2}\frac{\partial \rho_j}{\partial
x_i}(p_0)\eta_j\big) \\
&\quad + \sum_{k \in \mathcal{N}}
\frac{1}{\beta}\sum_{j=1}^n(x_{k}-x_{k}^0)\frac{2}{\rho_j(p_0)}
(\frac{\partial \rho_{k}}{\partial x_j}(p_0))^{-1}X_{jk}
+o(|t-t_0|+d_C(p_0,p)^2).
\end{align*}
Here, as in \cite{B:GS}, $\beta$ is the number of non-zero terms in the
final sum and we understand that if $\rho_j(p_0)=0$ or
$\frac{\partial \rho_{i_m}}{\partial x_j}(p_0)=0$ then that term in the
final sum is zero.
We define the subjet $P^{2,-}u(p_0,t_0)$ by
$$
P^{2,-}u(p_0,t_0)=-P^{2,+}(-u)(p_0,t_0).
$$
We also define the set theoretic closure of the superjet, denoted
$\overline{P}^{2,+}u(p_0,t_0)$, by requiring
$(a,\eta,X) \in \overline{P}^{2,+}u(p_0,t_0)$ exactly when there is a sequence
$$
(a_n,p_n,t_n,u(p_n,t_n),\eta_n,X_n)\to (a,p_0,t_0,u(p_0,t_0),\eta,X)
$$
with the triple $(a_n,\eta_n,X_n)\in P^{2,+}u(p_n,t_n)$. A similar
definition holds for the closure of the subjet.
As in the subelliptic case, we may also define jets using the appropriate
test functions. Namely, we consider the set $\mathcal{A}u(p_0,t_0)$ by
\[
\mathcal{A}u(p_0,t_0)=\{\phi \in C^2_{\rm sub}(\mathcal{O}_T): u(p,t)
-\phi(p,t) \leq u(p_0,t_0)-\phi(p_0,t_0)=0\}
\]
consisting of all test functions that touch from above. We define
the set of all test functions that touch from below, denoted
$\mathcal{B}u(p_0,t_0)$,
by
\[
\mathcal{B}u(p_0,t_0)=\{\phi \in C^2_{\rm sub}(\mathcal{O}_T):
u(p,t)-\phi(p,t) \geq u(p_0,t_0)-\phi(p_0,t_0)=0\}.
\]
The following lemma is proved in the same way as the Euclidean
version (\cite{C:VS} and \cite{I:I}) except we replace the Euclidean
distance $|p-p_0|$ with the local Grushin gauge $\mathcal{N}(p_0,p)$.
\begin{lemma} With the above notation, we have
\[
P^{2,+}u(p_0,t_0)=\{(\phi_t(p_0,t_0),\nabla_0 \phi(p_0,t_0),
(D^2\phi(p_0,t_0))^\star): \phi \in \mathcal{A}u(p_0,t_0)\}
\]
and
\[
P^{2,-}u(p_0,t_0)=\{(\phi_t(p_0,t_0),\nabla_0 \phi(p_0,t_0),
(D^2\phi(p_0,t_0))^\star): \phi \in \mathcal{B}u(p_0,t_0)\}.
\]
\end{lemma}
We may now relate the traditional Euclidean parabolic jets found
in \cite{CIL:UGTVS} to the Grushin subparabolic jets via the following lemma.
\begin{lemma} \label{jets}
Let the coordinates of the points $p,p_0 \in \mathbb{R}^n$ be
$p=(x_1,x_2,\dots,x_n)$ and $p_0=(x_1^0,x_2^0,\dots, x_n^0)$.
Let $P_{\rm eucl}^{2,+}u(p_0,t_0)$ be the traditional
Euclidean parabolic superjet of $u$ at the point $(p_0,t_0)$ and
let $ (a,\eta, X) \in \mathbb{R}\times \mathbb{R}^{n} \times
S^{n}$ with $\eta=(\eta_1,\eta_2,\dots,\eta_n)$. Then
$$
(a,\eta, X) \in \overline{P}_{\rm eucl}^{2,+}u(p_0,t_0)
$$
gives the element
$$
(a,\tilde{\eta},\mathcal{X}) \in \overline{P}^{2,+}u(p_0,t_0)
$$
where the vector $\tilde{\eta}$ is defined by
$$
\tilde{\eta} = \sum_{i=1}^n \rho_i(p_0) \eta_i X_i
$$
and the symmetric matrix $\mathcal{X}$ is defined by
\[
\mathcal{X}_{ij}=\begin{cases}
\rho_i(p_0)\rho_j(p_0)X_{ij}+\frac{1}{2}\frac{\partial \rho_j}{\partial
x_i}(p_0)\rho_i(p_0)\eta_j &\text{if } i \leq j \\
\mathcal{X}_{ji} &\text{if } i > j.
\end{cases}
\]
\end{lemma}
The proof matches the subelliptic case in Grushin-type spaces as
found in \cite{B:GS}.
We then use these jets to define subsolutions and supersolutions to
Equation \eqref{main}.
\begin{definition} \rm
Let $(p_0,t_0)\in \mathcal{O}_T$ be as above. The upper semicontinuous
function $u$ is a \emph{viscosity subsolution} in $\mathcal{O}_T$
if for all $(p_0,t_0) \in \mathcal{O}_T$ we have
$(a,\eta,X) \in P^{2,+}u(p_0,t_0)$ produces
\begin{equation}\label{sub}
a+F(t_0,p_0,u(p_0,t_0),\eta,X)\leq 0.
\end{equation}
A lower semicontinuous function $u$ is a \emph{viscosity supersolution} in
$\mathcal{O}_T$ if for all $(p_0,t_0) \in \mathcal{O}_T$ we have
$(b,\nu,Y) \in P^{2,-}u(p_0,t_0)$ produces
\begin{equation}\label{super}
b+F(t_0,p_0,u(p_0,t_0),\nu,Y)\geq 0.
\end{equation}
A continuous function $u$ is a \emph{viscosity solution} in
$\mathcal{O}_T$ if it is both a viscosity subsolution and
viscosity supersolution.
\end{definition}
We observe that the continuity of the function $F$ allows
Equations \eqref{sub} and \eqref{super} to hold when
$(a,\eta,X) \in \overline{P}^{2,+}u(p_0,t_0)$ and
$(b,\nu,Y) \in \overline{P}^{2,-}u(p_0,t_0)$, respectively.
We also wish to define what \cite{Ju:P} refers to as parabolic
viscosity solutions. We first need to consider the sets
$$
\mathcal{A}^-u(p_0,t_0)=\{\phi \in C^2_{\rm sub}(\mathcal{O}_T):
u(p,t)-\phi(p,t) \leq u(p_0,t_0)-\phi(p_0,t_0)=0 \text{ for }
t < t_0\}
$$
consisting of all functions that touch from above only when $t0$,
there are elements
$(a,\tau \Upsilon_{p_{\tau}},\mathcal{X}^{\tau}) \in
\overline{P}^{2,+}u(p_\tau,t_\tau)$ and
$(a,\tau \Upsilon{q_{\tau}},\mathcal{Y}^{\tau}) \in
\overline{P}^{2,-}v(q_\tau,t_\tau)$
where
\begin{gather*}
(\Upsilon_{p_\tau})_i \equiv \rho_i(p_\tau)\frac{\partial
\varphi(p_\tau, q_\tau)}{\partial x_i}
= \rho_i(p_\tau)(x^{\tau}_i-y^{\tau}_i)^{2^i-1}, \\
(\Upsilon_{q_\tau})_i \equiv -\rho_i(q_\tau)\frac{\partial
\varphi(p_\tau, q_\tau)}{\partial y_i} = \rho_i(q_\tau)(x^{\tau}_i-y^{\tau}_i)^{2^i-1}
\end{gather*}
so that if
$$
\lim_{\tau \to \infty}\tau\varphi(p_\tau,q_\tau)=0,
$$
then we have
\begin{gather}\label {vectordiff}
|\,\|\Upsilon_{q_\tau}\|^2-
\|\Upsilon_{p_\tau}\|^2\,| = O(\varphi(p_\tau,q_\tau)^2),\\
\label{matrixest}
\mathcal{X}^{\tau}\leq \mathcal{Y}^{\tau} +\mathcal{R}^{\tau}
\textmd{\; where\ }\lim_{\tau\to \infty}\mathcal{R}^{\tau}=0.
\end{gather}
We note that Equation \eqref{matrixest} uses the usual ordering
of symmetric matrices.
\end{theorem}
\begin{proof}
We first need to check that condition 8.5 of \cite{CIL:UGTVS} is satisfied,
namely that there exists an $r>0$ so that for each $M$, there exists a $C$
so that $b \leq C$ when $(b,\eta,X) \in P_{\rm eucl}^{2,+}u(p,t), |p-p_\tau|+|t-t_\tau|0$, we have an $M$ so that for all $C$, $b>C$ when $(b,\eta,X) \in P_{\rm eucl}^{2,+}u(p,t)$. By Lemma \ref{jets} we would have $$(b,\tilde{\eta},\mathcal{X})\in P^{2,+}u(p,t)$$
contradicting the fact that $u$ is a subsolution.
A similar conclusion is reached for $-v$ and so we conclude that this
condition holds. We may then apply Theorem 8.3 of \cite{CIL:UGTVS}
and obtain, by our choice of $\varphi$,
\begin{gather*}
(a,\tau D_p\varphi(p_\tau,q_\tau),X^\tau) \in
\overline{P}^{2,+}_{\rm eucl}u(p_\tau,t_\tau), \\
(a,-\tau D_q\varphi(p_\tau,q_\tau),Y^\tau) \in
\overline{P}^{2,-}_{\rm eucl}v(q_\tau,t_\tau).
\end{gather*}
Using Lemma \ref{jets} we define the vectors
$\Upsilon_{p_{\tau}}(p_\tau,q_\tau)$ and
$\Upsilon_{q_{\tau}}(p_\tau,q_\tau)$ by
\begin{gather*}
\Upsilon_{p_{\tau}}(p_\tau,q_\tau) = \widetilde{D_p\varphi}(p_\tau,q_\tau),\\
\Upsilon_{q_{\tau}}(p_\tau,q_\tau)
= -\widetilde{D_q\varphi}(p_\tau,q_\tau)
\end{gather*}
and we also define the matrices
$\mathcal{X}$ and $\mathcal{Y}$ as in Lemma \ref{jets}.
Then by Lemma \ref{jets},
\begin{gather*}
(a,\tau \Upsilon_{p_{\tau}}(p_\tau,q_\tau),\mathcal{X}^\tau)
\in \overline{P}^{2,+}u(p_\tau,t_\tau), \\
(a,\tau \Upsilon_{q_{\tau}}(p_\tau,q_\tau),\mathcal{Y}^\tau)
\in \overline{P}^{2,-}v(q_\tau,t_\tau).
\end{gather*}
Equations \eqref{vectordiff} and \eqref{matrixest} are
in \cite[Lemma 4.2]{B:GS}.
\end{proof}
Using this theorem, we now define a class of parabolic equations to
which we shall prove a comparison principle.
\begin{definition} \rm
We say the continuous, proper function
$$
F:[0,T]\times \overline{\Omega}\times \mathbb{R}\times g_n \times S^{n}
\to \mathbb{R}
$$
is \emph{admissible} if
for each $t \in [0,T]$, there is the same function
$\omega:[0,\infty] \to [0,\infty]$ with $\omega(0+)=0$ so that $F$ satisfies
\begin{equation}\label{cond}
F(t,q,r,\nu,\mathcal{Y})-F(t,p,r,\eta,\mathcal{X})
\leq\omega\big(d_C(p,q)+\big|\;\|\nu\|^2-\|\eta\|^2\big|+\|\mathcal{Y}-\mathcal{X}\|\big).
\end{equation}
\end{definition}
We now formulate the comparison principle for the following problem.
\begin{gather} %\label{problem}
u_t+F(t,p,u,\nabla_0 u, (D^2u)^\star) = 0 \quad
\textmd{in } (0,T)\times \Omega \label{E}\\
u(p,t)=h(p,t) \quad p \in \partial \Omega,\ t \in [0,T) \label{BC}\\
u(p,0) = \psi(p) \quad p \in \overline{\Omega} \label{IC}
\end{gather}
Here, $\psi \in C(\overline{\Omega})$ and
$h \in C(\overline{\Omega} \times [0,T))$.
We also adopt the convention in \cite{CIL:UGTVS}
that a subsolution $u(p,t)$ to Problem \eqref{E}--\eqref{IC} is a
viscosity subsolution to \eqref{E}, $u(p,t) \leq h(p,t)$ on
$\partial \Omega$ with $0 \leq t < T$ and $u(p,0) \leq \psi(p)$ on
$\overline{\Omega}$. Supersolutions and solutions are defined in
an analogous matter.
\begin{theorem}\label{comp}
Let $\Omega$ be a bounded domain in $G_n$. Let $F$ be admissible.
If $u$ is a viscosity subsolution and $v$ a viscosity supersolution to
Problem \eqref{E}--\eqref{IC} then $u \leq v$ on $[0,T) \times \Omega$.
\end{theorem}
\begin{proof}
Our proof follows that of \cite[Thm. 8.2]{CIL:UGTVS} and so we
discuss only the main parts.
For $\epsilon > 0$, we substitute $\tilde{u}=u-\frac{\varepsilon}{T-t}$
for $u$ and prove the theorem for
\begin{gather*}
u_t+F(t,p,u,\nabla_0 u,(D^2u)^\star) \leq -\frac{\varepsilon}{T^2} < 0, \\
\lim_{t \uparrow T}u(p,t) = -\infty \quad \text{uniformly on }
\overline{\Omega}
\end{gather*}
and take limits to obtain the desired result.
Assume the maximum occurs at $(p_0,t_0)\in \Omega \times (0,T)$
with
$$
u(p_0,t_0)-v(p_0,t_0)= \delta >0.
$$
Let
$$
M_\tau=u(p_\tau,t_\tau)-v(q_\tau,t_\tau)-\tau\varphi(p_\tau,q_\tau)
$$
with $(p_\tau,q_\tau,t_\tau)$ the maximum point in
$\overline{\Omega} \times \overline{\Omega} \times [0,T)$ of
$u(p,t)-v(q,t)-\tau \varphi(p,q)$.
Using the same proof as \cite[Lemma 5.2 ]{B:HG} we conclude that
$$
\lim_{\tau\to \infty}\tau\varphi(p_\tau,q_\tau) =0.
$$
If $t_\tau=0$, we have
$$
0 < \delta \leq M_\tau \leq
\sup_{\overline{\Omega}\times\overline{\Omega}}(\psi(p)-\psi(q)
-\tau \varphi(p,q))
$$
leading to a contradiction for large $\tau$. We therefore conclude
$t_\tau >0$ for large $\tau$. Since $u \leq v$ on
$\partial \Omega \times [0,T)$ by Equation \eqref{BC}, we conclude that
for large $\tau$, we have $(p_\tau,q_\tau,t_\tau)$ is an interior point.
That is,
$(p_\tau,q_\tau,t_\tau) \in \Omega \times \Omega \times (0,T)$.
Using Lemma \ref{jets}, we obtain
\begin{gather*}
(a,\tau \Upsilon_{p_{\tau}}(p_\tau,q_\tau), \mathcal{X}^\tau) \in
\overline{P}^{2,+}u(p_\tau,t_\tau), \\
(a,\tau \Upsilon_{q_{\tau}}(p_\tau,q_\tau), \mathcal{Y}^\tau) \in
\overline{P}^{2,-}v(q_\tau,t_\tau)
\end{gather*}
satisfying the equations
\begin{gather*}
a+F(t_\tau,p_\tau,u(p_\tau,t_\tau),\tau\Upsilon(p_\tau,q_\tau),
\mathcal{X}^\tau) \leq -\frac{\varepsilon}{T^2}, \\
a+F(t_\tau,q_\tau,v(q_\tau,t_\tau),\tau\Upsilon(p_\tau,q_\tau),
\mathcal{Y}^\tau) \geq 0.
\end{gather*}
Using the fact that $F$ is proper, the fact that
$u(p_\tau,t_\tau)\geq v(q_\tau,t_\tau)$ (otherwise $M_\tau < 0$),
and Equations \eqref{vectordiff} and \eqref{matrixest}, we have
\begin{align*}
0 0
$$
for a small parameter $\epsilon$.
Let $r > 0$ be sufficiently small so that the gauge
$\mathcal{N}(p_0,p)$ is comparable to the distance $d_C(p_0,p)$.
Define the gauge ball $B_{\mathcal{N}(p_0)}(r)$ by
$$
B_{\mathcal{N}(p_0)}(r)=\{p\in G_n:\mathcal{N}(p_0,p) \tilde{\phi}(p_0,t_0)$. Thus, the comparison principle,
Theorem \ref{comp}, does not hold. Thus, $u$ is not a viscosity subsolution.
The supersolution case is identical and omitted.
\end{proof}
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\end{document}